The real practical question is "Where does enough water start flowing together to show on this scale of map?" Because you're not going to want to map where every single drop of snowmelt starts its journey all the way to the sea. For every mapped river, there are all sorts of streams and rivulets and brooks and trickles that don't show. If you don't want to map anything that can be crossed by wading unless there's a flood, you could be into the lowlands before you need to start showing rivers. While some rivers begin in springs, others come out of glaciers, others are the run off of precipitation, either freshly-fallen or dumped as snow and slowly melting. Big rivers may trace back to all three. Lakes are not in themselves sources of rivers, but areas where water flows in and then out again. Where is the lake getting its water? Probably rivers or streams. Which is not to say that a mountain lake might not be the first place that the water shows on the map.

Of course, in fantasy mapping, there are additional sources of water possible. But wherever the water comes from, it's going to be at a high point compared to all the water downstream.

Anyway, my point is that you have decisions to make about how much river you want to show, which is going to affect where you want to start them. Look at a bunch of different continental maps. You should be able to find a good deal of variety in how much they show of the river systems. In terms of your maps, there's a good deal of wiggle room in how you show water flow.